Thread: Deriving a relationship between two angles

1. Deriving a relationship between two angles

Hey,

I am trying to derive a relationship between two angles.

The Problem: If you had a circular disk, and you removed a wedge from that disk then reconnected to two edges where the wedge had been removed from to create a cone, positioned that cone on a flat surface and measured the angle between the flat surface and the the wall of the cone to the vetex. What relationship would that angle have with the angle of the wedge which was initially removed?

Can this relationship be mathematically derived, without experimental data?

Any assistance would be greatly appreciated.

2. re: Deriving a relationship between two angles

From the explanation I would say as the arc length of the wedge removed increases then so does the angle.

What happens to the angle when the arc length exceeds half the circumference? Is this a restriction you need to consider?

3. Re: Deriving a relationship between two angles

Hello, Bemcma13!

You have a circular disk and you removed a wedge from that disk.
Then reconnected to two edges where the wedge had been removed
to create a cone, positioned that cone on a flat surface and measured
the angle between the flat surface and the the wall of the cone to the vetex.

What relationship would that angle have with the angle of the wedge
which was initially removed?

We have a circular disk of radius $R$.
We remove a sector with central angle $\theta.$

Code:
              * * *
*           *
*               *
*                 *

*                   *
*         *         *
*        /@\        *
R /:::\ R
*     /:::::\     *
*   /:::::::\   *
* :::::::::*
* * *

The length of the major arc is: $(2\pi-\theta)R$
This is the circumference of the circular base of the cone.

The side view of the cone looks like this:

Code:
              *
/|\
/ | \
/  |  \
R /   |   \ R
/    |    \
/     |     \
/      |    α \ π-α
*-------+-------*------
: - r - :

The slant height is $R.$
The radius of the base is $r.$
The base angle is $\alpha.$
You seek its supplement: $\pi-\alpha$

The circumference of the base circle is: $(2\pi - \theta)R$

We have: . $2\pi r \:=\:(2\pi-\theta)R \quad\Rightarrow\quad r \:=\:\frac{(2\pi-\theta)R}{2\pi}$

. . Then: . $\cos\alpha \:=\:\frac{r}{R} \:=\:\frac{2\pi-\theta}{2\pi} \:=\:1 - \frac{\theta}{2\pi}$

. . Hence: . $\alpha \:=\:\cos^{-1}\!\left(1 - \tfrac{\theta}{2\pi}\right)$

Therefore: . $\pi - \alpha \;=\;\pi - \cos^{-1}\!\left(1 -\tfrac{\theta}{2\pi}\right)$